Global Depths for Irregularly Observed Multivariate Functional Data

TL;DR

This paper introduces two new frameworks for multivariate functional depth, enhancing outlier detection and visualization in irregularly observed multivariate functional data, with proven properties and practical applications.

Contribution

It presents novel multivariate functional depth frameworks, including integrated and extremal depths, with theoretical properties, estimation methods, and visualization tools for irregular data.

Findings

Global depths outperform local depths in outlier detection.

Proposed methods are computationally efficient for large datasets.

Frameworks successfully applied to cyclone track data.

Abstract

Two frameworks for multivariate functional depth based on multivariate depths are introduced in this paper. The first framework is multivariate functional integrated depth, and the second framework involves multivariate functional extremal depth, which is an extension of the extremal depth for univariate functional data. In each framework, global and local multivariate functional depths are proposed. The properties of population multivariate functional depths and consistency of finite sample depths to their population versions are established. In addition, finite sample depths under irregularly observed time grids are estimated. As a by-product, the simplified sparse functional boxplot and simplified intensity sparse functional boxplot are proposed for visualization without data reconstruction. A simulation study demonstrates the advantages of global multivariate functional depths over local multivariate functional depths in outlier detection and running time for big functional data. An application of our frameworks to cyclone tracks data demonstrates the excellent performance of our global multivariate functional depths.